The graph of a line in the xy-plane has slope 2 and contains the point (1, 8). The graph of a second line passes through the points (1, 2) and (2, 1). If the two lines intersect at the point (a, b), what is the value of `a+b` ?

Options

A. 4
B. 3
C. -1
D. -4

Explanation

Choice B is correct. Since the slope of the first line is 2, an equation of this line can be written in the form y = 2x + c, where c is the y-intercept of the line. Since the line contains the point (1, 8), one can substitute 1 for x and 8 for y in y = 2x + c, which gives 8 = 2(1) + c, or c = 6. Thus, an equation of the first line is y = 2x + 6. The slope of the second line is equal to `(1 − 2)/(2 − 1)` or −1. Thus, an equation of the second line can be written in the form y = −x + d, where d is the y-intercept of the line. Substituting 2 for x and 1 for y gives 1 = −2 + d, or d = 3. Thus, an equation of the second line is y = −x + 3.
Since a is the x-coordinate and b is the y-coordinate of the intersection point of the two lines, one can substitute a for x and b for y in the two equations, giving the system b = 2a + 6 and b = –a + 3. Thus, a can be found by solving the equation 2a + 6 = −a + 3, which gives a = −1. Finally, substituting −1 for a into the equation b = –a + 3 gives b = −(−1) + 3, or b = 4. Therefore, the value of a + b is 3.
Alternatively, since the second line passes through the points (1, 2) and (2, 1), an equation for the second line is x + y = 3. Thus, the intersection point of the first line and the second line, (a, b) lies on the line with equation x + y = 3. It follows that a + b = 3.
Choices A and C are incorrect and may result from finding the value of only a or b, but not calculating the value of a + b. Choice D is incorrect and may result from a computation error in finding equations of the two lines or in solving the resulting system of equations.